23 research outputs found
On the stability of periodic 2D Euler-alpha flows
An explicit expression is obtained for the sectional curvature in the plane
spanned by two stationary flows, cos(k, x) and cos(l, x). It is shown that for
certain values of the wave vectors k and l the curvature becomes positive for
alpha > alpha_0, where 0 < alpha_0 < 1 is of the order 1/k. This suggests that
the flow corresponding to such geodesics becomes more stable as one goes from
usual Eulerian description to the Euler-alpha model
Abstract mechanical connection and Abelian reconstruction for almost Kähler manifolds
When the phase space P of a Hamiltonian G-system (P,ω,G,J,H) has an almost Kähler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form.
A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (see A. Blaom, Reconstruction
phases via Poisson reduction. Diff. Geom. Appl., 12:231{252, 2000.), thus signicantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure
Point vortices on a sphere: Stability of relative equilibria
In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle
Symmetry reduction of discrete Lagrangian mechanics on Lie groups
For a discrete mechanical system on a Lie group determined by a (reduced)
Lagrangian we define a Poisson structure via the pull-back of the
Lie-Poisson structure on the dual of the Lie algebra by the
corresponding Legendre transform. The main result shown in this paper is that
this structure coincides with the reduction under the symmetry group of the
canonical discrete Lagrange 2-form on . Its
symplectic leaves then become dynamically invariant manifolds for the reduced
discrete system. Links between our approach and that of groupoids and
algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid
body is discussed as an example
Discrete Euler-Poincar\'{e} and Lie-Poisson Equations
In this paper, discrete analogues of Euler-Poincar\'{e} and Lie-Poisson
reduction theory are developed for systems on finite dimensional Lie groups
with Lagrangians that are -invariant. These discrete
equations provide ``reduced'' numerical algorithms which manifestly preserve
the symplectic structure. The manifold is used as an approximation
of , and a discrete Langragian is
construced in such a way that the -invariance property is preserved.
Reduction by results in new ``variational'' principle for the reduced
Lagrangian , and provides the discrete
Euler-Poincar\'{e} (DEP) equations. Reconstruction of these equations recovers
the discrete Euler-Lagrange equations developed in \cite{MPS,WM} which are
naturally symplectic-momentum algorithms. Furthermore, the solution of the DEP
algorithm immediately leads to a discrete Lie-Poisson (DLP) algorithm. It is
shown that when , the DEP and DLP algorithms for a particular
choice of the discrete Lagrangian are equivalent to the
Moser-Veselov scheme for the generalized rigid body. %As an application, a
reduced symplectic integrator for two dimensional %hydrodynamics is constructed
using the SU approximation to the volume %preserving diffeomorphism group
of
Poisson structure and invariant manifolds on Lie groups
For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian ℓ we define a Poisson structure via the pull-back of the Lie-Poisson structure on g^∗ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω_L on G × G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system
Variational methods, multisymplectic geometry and continuum mechanics
This paper presents a variational and multisymplectic formulation of both
compressible and incompressible models of continuum mechanics on general
Riemannian manifolds. A general formalism is developed for non-relativistic
first-order multisymplectic field theories with constraints, such as the
incompressibility constraint. The results obtained in this paper set the stage
for multisymplectic reduction and for the further development of Veselov-type
multisymplectic discretizations and numerical algorithms. The latter will be
the subject of a companion paper